Chapter 2. Solving a Nonlinear Equation

Transcription

1 Chpter Soving Noniner Eqtion 1 Bisection Method Ater reding this chpter, yo shod be be to: 1 oow the gorith o the bisection ethod o soving noniner eqtion, se the bisection ethod to sove epes o inding roots o noniner eqtion, nd enerte the dvntges nd disdvntges o the bisection ethod One o the irst neric ethods deveoped to ind the root o noniner eqtion ( ) 0 ws the bisection ethod (so ced binry-serch ethod) The ethod is bsed on the oowing theore Theore An eqtion ( ) 0, where () is re continos nction, hs t est one root between i ( ) ( ) < 0 (See Figre 1) Note tht i ( ) ( ) > 0, there y or y not be ny root between nd (Figres nd ) I ( ) ( ) < 0, then there y be ore thn one root between (Figre 4) So the theore ony grntees one root between Bisection ethod Since the ethod is bsed on inding the root between two points, the ethod s nder the ctegory o brcketing ethods Since the root is brcketed between two points,, one cn ind the idpoint, between This gives s two intervs 1, nd () Figre 1 At est one root eists between the two points i the nction is re, continos, nd chnges sign 001

2 001 Chpter 001 () Figre I the nction () does not chnge sign between the two points, roots o the eqtion ( ) 0 y sti eist between the two points () () Figre I the nction () does not chnge sign between two points, there y not be ny roots or the eqtion ( ) 0 between the two points

3 Bisection Method 001 () Figre 4 I the nction () chnges sign between the two points, ore thn one root or the eqtion ( ) 0 y eist between the two points Is the root now between or between? We, one cn ind the sign o ( ) ( ), nd i ( ) ( ) < 0 then the brcket is between, otherwise, it is between So, yo cn see tht yo re itery hving the interv As one repets this process, the width o the interv [, ] becoes ser nd ser, nd yo cn zero in to the root o the eqtion ( ) 0 The gorith or the bisection ethod is given s oows Agorith or the bisection ethod The steps to ppy the bisection ethod to ind the root o the eqtion ( ) 0 re 1 Choose s two gesses or the root sch tht ( ) ( ) < 0, or in other words, () chnges sign between Estite the root,, o the eqtion ( ) 0 s the id-point between nd s + Now check the oowing ) I ( ) ( ) < 0, then the root ies between ; then nd b) I ( ) ( ) > 0, then the root ies between ; then nd c) I ( ) ( ) 0 ; then the root is Stop the gorith i this is tre 4 Find the estite o the root + Find the bsote retive pproite error s

4 0014 Chpter 001 where - od estited root ro present itertion estited root ro previos itertion od 5 Copre the bsote retive pproite error with the pre-speciied Epe 1 retive error toernce s I >s, then go to Step, ese stop the gorith Note one shod so check whether the nber o itertions is ore thn the i nber o itertions owed I so, one needs to terinte the gorith nd notiy the ser bot it Yo re working or DOWN THE TOILET COMPANY tht kes ots or ABC coodes The oting b hs speciic grvity o 06 nd hs rdis o 55 c Yo re sked to ind the depth to which the b is sberged when oting in wter The eqtion tht gives the depth to which the b is sberged nder wter is given by Use the bisection ethod o inding roots o eqtions to ind the depth to which the b is sberged nder wter Condct three itertions to estite the root o the bove eqtion Find the bsote retive pproite error t the end o ech itertion, nd the nber o signiicnt digits t est correct t the end o ech itertion Sotion Fro the physics o the probe, the b wod be sberged between 0 R, where R rdis o the b, tht is 0 R 0 (0055) Figre 5 Foting b probe

5 Bisection Method 0015 Lets s sse 0, 011 Check i the nction chnges sign between ( ) (0) (0) 0165(0) Hence ( ( ) (011) (011) ) ( ) 0165(011) (0) (011) ( )( ) < 0 So there is t est one root between, tht is between 0 nd 011 Itertion 1 The estite o the root is ( ) ( ) (0) (0055) ( 99 )( 6655 ) > 0 Hence the root is brcketed between, tht is, between 0055 nd 011 So, the ower nd pper iit o the brcket is 0 055, 011 At this point, the bsote retive pproite error not hve previos pproition Itertion The estite o the root is ( ) (0085) (0085) 0165(0085) cnnot be ccted s we do ( ) ( ) ( 0055) ( 0085) ( 6655 ) ( 16 ) < 0 Hence, the root is brcketed between, tht is, between 0055 nd 0085 So the ower nd pper iit o the brcket is 0 055, 0085 The bsote retive pproite error ( ) ( ) ( 0055) 0165( 0055) od % t the end o Itertion is

6 0016 Chpter 001 None o the signiicnt digits re t est correct in the estited root o becse the bsote retive pproite error is greter thn 5% Itertion ( ) (006875) (006875) 0165(006875) ( ) ( ) 5 (0055) (006875) (6655 ) ( 56 ) < 0 Hence, the root is brcketed between, tht is, between 0055 nd So the ower nd pper iit o the brcket is 0 055, The bsote retive pproite error od t the ends o Itertion is % Sti none o the signiicnt digits re t est correct in the estited root o the eqtion s the bsote retive pproite error is greter thn 5% Seven ore itertions were condcted nd these itertions re shown in Tbe 1 Tbe 1 Root o ( ) 0 s nction o nber o itertions or bisection ethod Itertion % ( ) At the end o th itertion, 0171% Hence the nber o signiicnt digits t est correct is given by the rgest ve o or which og( 044)

7 Bisection Method 0017 og(044) 46 So The nber o signiicnt digits t est correct in the estited root o t the th end o the itertion is Advntges o bisection ethod ) The bisection ethod is wys convergent Since the ethod brckets the root, the ethod is grnteed to converge b) As itertions re condcted, the interv gets hved So one cn grntee the error in the sotion o the eqtion Drwbcks o bisection ethod ) The convergence o the bisection ethod is sow s it is sipy bsed on hving the interv b) I one o the initi gesses is coser to the root, it wi tke rger nber o itertions to rech the root c) I nction () is sch tht it jst toches the -is (Figre 6) sch s ( ) 0 it wi be nbe to ind the ower gess,, nd pper gess,, sch tht ( ) ( ) < 0 d) For nctions () where there is singrity 1 nd it reverses sign t the singrity, the bisection ethod y converge on the singrity (Figre 7) An epe incdes 1 ( ) where, re vid initi gesses which stisy ( ) ( ) < 0 However, the nction is not continos nd the theore tht root eists is so not ppicbe () Figre 6 The eqtion ( ) 0 hs singe root t 0 tht cnnot be brcketed

8 0018 Chpter A singrity in nction is deined s point where the nction becoes ininite For epe, or nction sch s 1 /, the point o singrity is 0 s it becoes ininite () Figre 7 The eqtion ( ) 0 1 hs no root bt chnges sign